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Round-rotor synchronous machine with fundamental parameterization
The Synchronous Machine Round Rotor (fundamental) block models a round-rotor synchronous machine with parameterization using fundamental parameters.
The synchronous machine equations are expressed with respect to a rotating reference frame defined by the equation
$${\theta}_{e}(t)=N*{\theta}_{r}(t),$$
where:
θ_{e} is the electrical angle.
N is the number of pole pairs.
θ_{r} is the rotor angle.
Park's transformation maps the synchronous machine equations to the rotating reference frame with respect to the electrical angle. Park's transformation is defined by
$${P}_{s}=\frac{2}{3}\left[\begin{array}{ccc}\mathrm{cos}{\theta}_{e}& \mathrm{cos}({\theta}_{e}-\frac{2\pi}{3})& \mathrm{cos}({\theta}_{e}+\frac{2\pi}{3})\\ -\mathrm{sin}{\theta}_{e}& -\mathrm{sin}({\theta}_{e}-\frac{2\pi}{3})& -\mathrm{sin}({\theta}_{e}-\frac{2\pi}{3})\\ \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\end{array}\right].$$
Park's transformation is used to define the per-unit synchronous machine equations. The stator voltage equations are defined by
$${e}_{d}=\frac{1}{{\omega}_{base}}\frac{\text{d}{\psi}_{d}}{\text{d}t}-{\Psi}_{q}{\omega}_{r}-{R}_{a}{i}_{d},$$
${e}_{q}=\frac{1}{{\omega}_{base}}\frac{\text{d}{\psi}_{q}}{\text{d}t}+{\Psi}_{d}{\omega}_{r}-{R}_{a}{i}_{q},$
and
$${e}_{0}=\frac{1}{{\omega}_{base}}\frac{d{\Psi}_{0}}{dt}-{R}_{a}{i}_{0},$$
where:
e_{d}, e_{q}, and e_{0} are the d-axis, q-axis, and zero-sequence stator voltages, defined by
$$\left[\begin{array}{c}{e}_{d}\\ {e}_{q}\\ {e}_{0}\end{array}\right]={P}_{s}\left[\begin{array}{c}{v}_{a}\\ {v}_{b}\\ {v}_{c}\end{array}\right],$$
where v_{a}, v_{b}, and v_{c} are the stator voltages measured from port ~ to neutral port n.
ω_{base} is the per-unit base electrical speed.
ψ_{d}, ψ_{q}, and ψ_{0} are the d-axis, q-axis, and zero-sequence stator flux linkages.
ω_{r} is the per-unit rotor rotational speed.
R_{a} is the stator resistance.
i_{d}, i_{q} and i_{0} are the d-axis, q-axis, and zero-sequence stator currents, defined by
$$\left[\begin{array}{c}{i}_{d}\\ {i}_{q}\\ {i}_{0}\end{array}\right]={P}_{s}\left[\begin{array}{c}{i}_{a}\\ {i}_{b}\\ {i}_{c}\end{array}\right],$$
where i_{a}, i_{b}, and i_{c} are the stator currents flowing from port ~ to port n.
The rotor voltage equations are defined by
$${e}_{fd}=\frac{1}{{\omega}_{base}}\frac{d{\Psi}_{fd}}{dt}+{R}_{fd}{i}_{fd},$$
$${e}_{1d}=\frac{1}{{\omega}_{base}}\frac{d{\Psi}_{1d}}{dt}+{R}_{1d}{i}_{1d}=0,$$
$${e}_{1}{}_{q}=\frac{1}{{\omega}_{base}}\frac{d{\Psi}_{1q}}{dt}+{R}_{1q}{i}_{1q}=0,$$
and
$${e}_{2}{}_{q}=\frac{1}{{\omega}_{base}}\frac{d{\Psi}_{2q}}{dt}+{R}_{2q}{i}_{2q}=0,$$
where:
e_{fd} is the field voltage.
e_{1d}, e_{1q}, and e_{2q} are the voltages across the d-axis damper winding 1, q-axis damper winding 1, and q-axis damper winding 2. They are all equal to 0.
ψ_{fd}, ψ_{1d}, ψ_{1q}, and ψ_{2q} are the magnetic fluxes linking the field circuit, d-axis damper winding 1, q-axis damper winding 1, and q-axis damper winding 2.
R_{fd}, R_{1d}, R_{1q}, and R_{2q} are the resistances of rotor field circuit, d-axis damper winding 1, q-axis damper winding 1, and q-axis damper winding 2.
i_{fd}, i_{1d}, i_{1q}, and i_{2q} are the currents flowing in the field circuit, d-axis damper winding 1, q-axis damper winding 1, and q-axis damper winding 2.
The saturation equations are defined by
$${\psi}_{at}=\sqrt{{\psi}_{d}^{2}+{\psi}_{q}^{2}},$$
$${K}_{s}=1$$ (If saturation is disabled),
${K}_{s}=f\left({\psi}_{at}\right)$ (If saturation is enabled),
${L}_{ad}={K}_{s}*{L}_{adu},$
and
${L}_{aq}={K}_{s}*{L}_{aqu},$
where:
ψ_{at} is the air-gap flux linkage.
K_{s} is the saturation factor.
L_{adu} is the unsaturated mutual inductance of the stator d-axis.
L_{ad} is the mutual inductance of the stator d-axis.
L_{aqu} is the unsaturated mutual inductance of the stator q-axis.
L_{aq} is the mutual inductance of the stator q-axis.
The saturation factor function, f, is calculated from the per-unit open-circuit lookup table as:
${L}_{ad}=\frac{d{\psi}_{at}}{d{i}_{fd}},$
${V}_{ag}=g({i}_{fd}),$
and
${L}_{ad}=\frac{dg({i}_{fd})}{d{i}_{fd}}=\frac{d{V}_{ag}}{d{i}_{fd}},$
where:
V_{ag} is the per-unit air-gap voltage.
In per-unit,
${K}_{s}=\frac{{L}_{ad}}{{L}_{adu}},$
and
${\psi}_{at}={V}_{ag}$
can be rearranged to
${K}_{s}=f({\psi}_{at}).$
The stator flux linkage equations are defined by
$${\Psi}_{d}=-({L}_{ad}+{L}_{i}){i}_{d}\text{}+{L}_{ad}{i}_{fd}+{L}_{ad}{i}_{1d},$$
$$\Psi q=-({L}_{aq}+{L}_{i}){i}_{q}\text{}+{L}_{aq}{i}_{1q}+{L}_{aq}{i}_{2q},$$
and
$${\Psi}_{0}=-{L}_{0}{i}_{0},$$
where:
L_{l} is the stator leakage inductance.
L_{ad} and L_{aq} are the mutual inductances of the stator d-axis and q-axis.
The rotor flux linkage equations are defined by
$${\psi}_{fd}={L}_{ffd}{i}_{fd}+{L}_{f1d}{i}_{1d}-{L}_{ad}{i}_{d},$$
$${\psi}_{1d}={L}_{f1d}{i}_{fd}+{L}_{11d}{i}_{1d}-{L}_{ad}{i}_{d},$$
$${\psi}_{1q}={L}_{11q}{i}_{1q}+{L}_{aq}{i}_{2q}-{L}_{aq}{i}_{q},$$
and
$${\psi}_{2q}={L}_{aq}{i}_{1q}+{L}_{22q}{i}_{2q}-{L}_{aq}{i}_{q},$$
where:
L_{ffd}, L_{11d}, L_{11q}, and L_{22q} are the self-inductances of the rotor field circuit, d-axis damper winding 1, q-axis damper winding 1 and q-axis damper winding 2. L_{f1d} is the rotor field circuit and d-axis damper winding 1 mutual inductance. They are defined by the following equations.
$${L}_{ffd}={L}_{ad}+{L}_{fd}$$
$${L}_{f1d}={L}_{ffd}-{L}_{fd}$$
$${L}_{11d}={L}_{f1d}+{L}_{1d}$$
$${L}_{11q}={L}_{aq}+{L}_{1q}$$
$${L}_{22q}={L}_{aq}+{L}_{2q}$$
These equations assume that per-unit mutual inductance L_{12q} = L_{aq}, i.e., the stator and rotor currents in the q-axis all link a single mutual flux represented by L_{aq}.
The rotor torque is defined by
$${T}_{e}={\Psi}_{d}{i}_{q}-{\Psi}_{q}{i}_{d}.$$
For synchronous machine blocks, you can perform display actions using the Power Systems menu on the block context menu.
Right-click the block. From the context menu, select one of the following from the Power Systems > Synchronous Machine menu:
Display Base Values displays the machine per-unit base values at the MATLAB^{®} command prompt.
Display Associated Base Values displays associated per-unit base values at the MATLAB command prompt.
Associated Initial Conditions displays associated initial conditions at the MATLAB command prompt.
Plot Open-Circuit Saturation (pu) plots air-gap voltage, V_{ag}, versus field current, i_{fd}, (both measured in per-unit) in a MATLAB figure window. The plot contains three traces:
Unsaturated: Stator d-axis mutual inductance (unsaturated), Ladu you specify
Saturated: Per-unit open-circuit lookup table (Vag versus ifd) you specify
Derived: Open-circuit lookup table (per-unit) derived from the Per-unit open-circuit lookup table (Vag versus ifd) you specify. This data is used to calculate the saturation factor, K_{s}, versus magnetic flux linkage, ψ_{at}, characteristic.
Plot Saturation Factor (pu) plots saturation factor, K_{s}, versus magnetic flux linkage, ψ_{at}, (both measured in per-unit) in a MATLAB figure window using the present machine parameters. This is derived from parameters you specify:
Stator d-axis mutual inductance (unsaturated), Ladu
Per-unit field current saturation data, ifd
Per-unit air-gap voltage saturation data, Vag
Rated apparent power. The default value is 555e6 VA.
RMS rated line-line voltage. The default value is 24e3 V.
Nominal electrical frequency at which rated apparent power is quoted. The default value is 60 Hz.
Number of machine pole pairs. The default value is 1.
Select between
Field circuit voltage
Field circuit current
The default value is Field circuit current.
This parameter is visible only when Specify field circuit input required to produce rated terminal voltage at no load by is set to Field circuit current. The default value is 1300 A.
This parameter is visible only when Specify field circuit input required to produce rated terminal voltage at no load by is set to Field circuit voltage. The default value is 92.95 V.
Unsaturated stator d-axis mutual inductance. If Magnetic saturation representation is set to None, this is equivalent to the stator d-axis mutual inductance. The default value is 1.66 pu.
Unsaturated stator q-axis mutual inductance. If Magnetic saturation representation is set to None, this is equivalent to the stator q-axis mutual inductance. The default value is 1.61 pu.
Stator zero-sequence inductance. The default value is 0 pu.
Stator leakage inductance. The default value is 0.15 pu.
Stator resistance. The default value is 0.003 pu.
Rotor field circuit inductance. The default value is 0.165 pu.
Rotor field circuit resistance. The default value is 0.0006 pu.
Rotor d-axis damper winding 1 inductance. The default value is 0.1713 pu.
Rotor d-axis damper winding 1 resistance. The default value is 0.0284 pu.
Rotor q-axis damper winding 1 inductance. The default value is 0.7252 pu.
Rotor q-axis damper winding 1 resistance. The default value is 0.00619 pu.
Rotor q-axis damper winding 2 inductance. The default value is 0.125 pu.
Rotor q-axis damper winding 2 resistance. The default value is 0.02368 pu.
Block magnetic saturation representation. Options are:
None
Per-unit open-circuit lookup table (Vag versus ifd)
The default value is None.
The field current, i_{fd}, data populates the air-gap voltage, V_{ag}, versus field current, i_{fd}, lookup table. This parameter is only visible when you set Magnetic saturation representation to Per-unit open-circuit lookup table (Vag versus ifd). This parameter must contain a vector with at least five elements. The default value is [0.00, 0.48, 0.76, 1.38, 1.79] pu.
The air-gap voltage, V_{ag}, data populates the air-gap voltage, V_{ag}, versus field current, i_{fd}, lookup table. This parameter is only visible when you set Magnetic saturation representation to Per-unit open-circuit lookup table (Vag versus ifd). This parameter must contain a vector with at least five elements. The default value is [0.00, 0.80, 1.08, 1.31, 1.40] pu.
Options include:
Electrical power and voltage output
Mechanical and magnetic states
The default value is Electrical power and voltage output.
Initial RMS line-line voltage. This parameter is visible only when you set Specify initialization by to Electrical power and voltage output. The default value is 24e3 V.
Initial voltage angle. This parameter is visible only when you set Specify initialization by to Electrical power and voltage output. The default value is 0 deg.
Initial active power. This parameter is visible only when you set Specify initialization by to Electrical power and voltage output. The default value is 500e6 VA.
Initial reactive power. This parameter is visible only when you set Specify initialization by to Electrical power and voltage output. The default value is 0 VA.
Initial rotor angle. During steady-state operation, set this parameter to the sum of the load angle and required terminal voltage offset. This parameter is visible only when you set Specify initialization by to Mechanical and magnetic states. The default value is 0 deg.
Stator d-axis initial flux linkage. This parameter is visible only when you set Specify initialization by to Mechanical and magnetic states. The default value is 0 pu.
Stator q-axis initial flux linkage. This parameter is visible only when you set Specify initialization by to Mechanical and magnetic states. The default value is 0 pu.
Zero-sequence initial flux linkage. This parameter is visible only when you set Specify initialization by to Mechanical and magnetic states. The default value is 0 pu.
Field circuit initial flux linkage. This parameter is visible only when you set Specify initialization by to Mechanical and magnetic states. The default value is 0 pu.
d-axis damper winding 1 initial flux linkage. This parameter is visible only when you set Specify initialization by to Mechanical and magnetic states. The default value is 0 pu.
The q-axis damper winding 1 initial flux linkage. This parameter is visible only when you set Specify initialization by to Mechanical and magnetic states. The default value is 0 pu.
The q-axis damper winding 2 initial flux linkage. This parameter is visible only when you set Specify initialization by to Mechanical and magnetic states. The default value is 0 pu.
The block has the following ports:
Electrical conserving port corresponding to the field winding positive terminal
Electrical conserving port corresponding to the field winding negative terminal
Mechanical rotational conserving port associated with the machine rotor
Mechanical rotational conserving port associated with the machine case
Physical signal vector port associated with the machine per-unit measurements. The vector elements are:
pu_fd_Efd
pu_fd_Ifd
pu_torque
pu_velocity
pu_ed
pu_ed
pu_e0
pu_id
pu_iq
pu_i0
Expandable three-phase port associated with the stator windings
Electrical conserving port associated with the neutral point of the wye winding configuration
[1] Kundur, P. Power System Stability and Control. New York, NY: McGraw Hill, 1993.
[2] Lyshevski, S. E. Electromechanical Systems, Electric Machines and Applied Mechatronics. Boca Raton, FL: CRC Press, 1999.